Effect of strong magnetic fields on the crust-core transition and inner crust of neutron stars

The Vlasov equation is used to determine the dispersion relation for the eigenmodes of magnetized nuclear and neutral stellar matter, taking into account the anomalous magnetic moment of nucleons. The formalism is applied to the determination of the dynamical spinodal section, a quantity that gives a good estimation of the crust-core transition in neutron stars. We study the effect of strong magnetic fields, of the order of ${10}^{15}–{10}^{17}$ G, on the extension of the crust of magnetized neutron stars. The dynamical instability region of neutron-proton-electron ($npe$) matter at subsaturation densities is determined within a relativistic mean-field model. It is shown that a strong magnetic field has a large effect on the instability region, defining the crust-core transition as a succession of stable and unstable regions due to the opening of new Landau levels. The effect of the anomalous magnetic moment is non-negligible for fields larger than ${10}^{15}$ G. The complexity of the crust at the transition to the core and the increase of the crust thickness may have direct impact on the properties of neutrons stars related with the crust.

Figure 1

Dynamical spinodal for $\mathrm{NL}3\omega \rho$, a momentum transfer of $k=75$ MeV, and $B^{*}={10}^4$ (red), $B^{*}={10}^3$ (blue), and $B^{*}={10}^2$ (green) with AMM (top) and without AMM (bottom). A comparison with the $B=0$ (black lines) results is also made.

Figure 2

Dynamical spinodal for $\mathrm{NL}3\omega \rho$, a momentum transfer of $k=75$ MeV, with AMM (green) and without AMM (red), for $B^{*}={10}^4$ (top), $B^{*}={10}^3$ (middle), and $B^{*}={10}^2$ (bottom). A comparison with the $B=0$ (black lines) results is also made. The EoS for $\beta$-equilibrium matter is also shown.

Figure 3

Growth rates $\left|\omega \right|$ as a function of the momentum $k$ for $\mathrm{NL}3\omega \rho$, a proton fraction of $y_p=0.035$, and $B^{*}={10}^2$ (red), $B^{*}={10}^3$ (black), and $B^{*}={10}^4$ (blue) with AMM (solid) and without AMM (dashed lines), for a fixed baryonic density of $\rho =0.04{\mathrm{fm}}^{-3}$ (top), $\rho =0.08{\mathrm{fm}}^{-3}$ (middle), and $\rho =0.09{\mathrm{fm}}^{-3}$ (bottom). The growth rates for $B^{*}={10}^2$ in the bottom panel are multiplied by a factor of 10 and are obtained for $\rho =0.0903{\mathrm{fm}}^{-3}$.

Figure 4

Largest growth rate (top panels), the corresponding half wavelength (middle panels), and the proton-neutron density fluctuation ratio (bottom panels) versus density for $\mathrm{NL}3\omega \rho$, and a magnetic field of $B^{*}={10}^3$, without (left panels) and with (right panels) AMM. Results with the $B=0$ (black solid lines) calculation are also shown for comparison. The blue points correspond to a calculation with a proton fraction equal to the one found in $\beta$-equilibrium matter, above ${\rho }_t=0.0843{\mathrm{fm}}^{-3}$, the $B=0$ crust-core transition density. The gray points correspond to a fixed proton fraction of 0.035 in the whole density range.

Figure 5

Largest growth rate (top panels), the corresponding half wavelength (middle panels), and the proton-neutron density fluctuation ratio (bottom panels) versus density for $\mathrm{NL}3\omega \rho$, with a proton fraction of $y_p=0.035$, and without (left panels) and with (right panels) AMM. Three different values of $B$ are considered: $B^{*}={10}^2$ (red), $B^{*}={10}^3$ (green), and $B^{*}={10}^4$ (blue). A comparison with the $B=0$ (black lines) results is also made.

Figure 6

Width of the instability peaks, $\mathrm{\Delta }\rho$ (top panel), the thickness of the instability regions, $\mathrm{\Delta }l$ (middle panel), and the distance between the instabilities, $\mathrm{\Delta }L$ (bottom panel), as a function of the baryonic density, for $\mathrm{NL}3\omega \rho$, a proton fraction of $y_p=0.035, B^{*}={10}^2$ (red), and $B^{*}={10}^3$ (blue), with AMM. In the bottom panel, $\mathrm{\Delta }L$ obtained with $B^{*}={10}^3$ is multiplied by a factor of 0.1.